Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{9q^2 + 117q + 360}{-5q^2 - 90q - 400}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {9(q^2 + 13q + 40)} {-5(q^2 + 18q + 80)} $ $ r = -\dfrac{9}{5} \cdot \dfrac{q^2 + 13q + 40}{q^2 + 18q + 80} $ Next factor the numerator and denominator. $ r = - \dfrac{9}{5} \cdot \dfrac{(q + 8)(q + 5)}{(q + 8)(q + 10)}$ Assuming $q \neq -8$ , we can cancel the $q + 8$ $ r = - \dfrac{9}{5} \cdot \dfrac{q + 5}{q + 10}$ Therefore: $ r = \dfrac{ -9(q + 5)}{ 5(q + 10)}$, $q \neq -8$